An Adaptive Sparse Haar-Transform Method Using the Fast Fourier Transform for Two-Dimensional Nonlinear Integral Equations: Stability, Error Control, and Numerical Validation

Abstract

     Two-dimensional nonlinear Hammerstein and Fredholm integral equations lead to globally coupled discrete operators, for which direct kernel evaluation requires quadratic arithmetic and storage. This study develops an adaptive sparse Haar–FFT (ASH-FFT) formulation for translation-invariant periodic kernels and for nonperiodic kernels that admit a zero-padded convolution embedding. The algorithm solves a nonlinear equation on an active Haar coefficient set, applying the nonlinear operator and Jacobian–vector products matrix-free on an embedding grid. Under the stated contraction condition , established fixed-point and perturbation arguments are specialized to obtain well-posedness, stability, and a computable residual error certificate; the algorithm-specific analysis then separates the embedding-grid size  from the active coefficient count . One residual or Jacobian–vector product costs  operations and uses  working memory; the reduced expression  applies only to graded/local implementations satisfying . In the reported tests, the piecewise-constant Haar approximation shows first-order behavior: the  error decreases from  to  as  increases from 1,024 to 65,536. A localized nonlinear test retains 10.16% of the coefficients with a relative error of 0.0215%, whereas a weakly singular test retains 82.86%, showing that sparsity is problem dependent. Operator-level FFT speedups over direct circular convolution range from 4.11 to  on the tested grids. Validation also includes a published two-dimensional Haar benchmark, perturbation tests, and a grid-refined neural-field example. The claims are therefore restricted to the stated contraction regime, FFT-embeddable kernels, and the tested grid range.